Let R be a primitive ring with nonzero socle, M a faithful irreducible right R-module, A the central-izer of M, and L= a direct sum of countably many minimal right ideals L, of R. Then there existsa family of subsetsi...Let R be a primitive ring with nonzero socle, M a faithful irreducible right R-module, A the central-izer of M, and L= a direct sum of countably many minimal right ideals L, of R. Then there existsa family of subsetsis infinite) of R such that L=R for any W, whereeach is a set of countably many orthogonal idempotent elements of rank one in R. Furthermore,there exists a primitive ring R and a direct sum L=of countably many minimal right ideals Li ofR, but R has no subset B =of countably many orthogonal idempotent elements of rank one such that and B can be extended to a corresponding basis of some basis of M over A.展开更多
The structure theorem of the primitive rings with non-zero socles has been proved in Refs. [1] and (2)In the former, the finite topological method is used; and in the latter the method of right and left double modules...The structure theorem of the primitive rings with non-zero socles has been proved in Refs. [1] and (2)In the former, the finite topological method is used; and in the latter the method of right and left double modules. It should be noted that there is a close relation between the structure of the primitive rings of this展开更多
It is well known that,for a subring of a full linear ring over a vector spaec,2-foldtransitive implies k-fold transitive for every natual integer k,and a primitive ring withminimal oneside ideal is a two side nonsingu...It is well known that,for a subring of a full linear ring over a vector spaec,2-foldtransitive implies k-fold transitive for every natual integer k,and a primitive ring withminimal oneside ideal is a two side nonsingular ring and every isomorphism can be inducedby a semi-linear one to one transformation.This paper generalizes these results to weaklyprimitive rings.展开更多
In this paper the concept △-context is introduced,as a generalization of MoritaContext,to give a characterization of nonsingularly relative primitive rings.
Let G be an arbitrary group, finite or infinite, and A be a G-graded ring, i. e. A is an associative ring and A<sub>g</sub> （direct sum of additive subgroups A<sub>g</sub> ）with property: A&l...Let G be an arbitrary group, finite or infinite, and A be a G-graded ring, i. e. A is an associative ring and A<sub>g</sub> （direct sum of additive subgroups A<sub>g</sub> ）with property: A<sub>g</sub>· A<sub>h</sub> A<sub>gh</sub>, g, h∈G. Let M be a graded A-module, i. e. M is right A-module and M<sub>g</sub>展开更多
Peal[2] shows that a sufficient and necessary condition on the existence of theMoore-Penrose inverse over any fields.Zhuang [3] generalize the result to any divisionrings.In this section we give another sufficient and...Peal[2] shows that a sufficient and necessary condition on the existence of theMoore-Penrose inverse over any fields.Zhuang [3] generalize the result to any divisionrings.In this section we give another sufficient and necessary condition on the existence ofthe Moore-Penrose inverse over any division rings.Our result can be regarded as an im-provement of Theorem lin[1].As a medium result,we also show a characterization ofthe{1,2}-inverse.展开更多
A certain variety of non-switched polynomials provides a uni-figure representation for a wide range of linear functional equations. This is properly adapted for the calculations. We reinterpret from this point of view...A certain variety of non-switched polynomials provides a uni-figure representation for a wide range of linear functional equations. This is properly adapted for the calculations. We reinterpret from this point of view a number of algorithms.展开更多
Let Z/（p^e） be the integer residue ring modulo p^e with p an odd prime and integer e ≥ 3. For a sequence a over Z/（p^e）, there is a unique p-adic decomposition a- = a-0 ＋a-1 .p ＋… ＋ a-e-l .p^e-1 where each a-...Let Z/（p^e） be the integer residue ring modulo p^e with p an odd prime and integer e ≥ 3. For a sequence a over Z/（p^e）, there is a unique p-adic decomposition a- = a-0 ＋a-1 .p ＋… ＋ a-e-l .p^e-1 where each a-i can be regarded as a sequence over Z/（p）, 0 ≤ i ≤ e - 1. Let f（x） be a primitive polynomial over Z/（p^e） and G＇（f（x）,p^e） the set of all primitive sequences generated by f（x） over Z/（p^e）. For μ（x） ∈ Z/（p）[x] with deg（μ（x）） ≥ 2 and gad（1 ＋ deg（μ（x））,p- 1） = 1, setφe-1 （x0, x1,… , xe-1） = xe-1. [μ（xe-2） ＋ ηe-3（x0, X1,…, xe-3）] ＋ ηe-2（x0, X1,…, xe-2） which is a function of e variables over Z/（p）. Then the compressing mapφe-1 ： G＇（f（x）,p^e） → （Z/（p））^∞ ,a-→φe-1（a-0,a-1, … ,a-e-1） is injective. That is, for a-,b-∈ G＇（f（x）,p^e）, a- = b- if and only if φe-1 （a-0,a-1, … ,a-e-1） = φe-1（b-0, b-1,… ,b-e-1）. As for the case of e = 2, similar result is also given. Furthermore, if functions φe-1 and ψe-1 over Z/（p） are both of the above form and satisfy φe-1（a-0,a-1,…,a-e-1）=ψe-1（b-0, b-1,… ,b-e-1） for a-,b-∈G＇（f（x）,p^e）, the relations between a- and b-, φe-1 and ψe-1 are discussed展开更多
The theory of primitive polynomials over Galois rings is analogue to the same one over finite fields. It also provides useful tools for one to study the maximal period sequences over Galois rings. In the case of F<...The theory of primitive polynomials over Galois rings is analogue to the same one over finite fields. It also provides useful tools for one to study the maximal period sequences over Galois rings. In the case of F<sub>q</sub>, we have more complete results. In the case of Z<sub>p<sup>n</sup></sub>, n≥2, there are also some results. In particular, according to refs. [3, 4] and using the technique of trace representation of maximal period sequences over F<sub>q</sub>, we have found a discriminant which can judge whether a given polynomial f（x） over Z<sub>p<sup>n</sup></sub> is a primitive polynomial if f（x） mod p is a primitive polynomial over F<sub>p</sub>. Furthermore, it is easy to calculate the discriminant using the coefficients of f（x）.展开更多
Many authors have studied the problem of determining the linear operators, on the n×n matrix algebra M<sub>n</sub>（R） over a commutative R, which preserve idempotent matrices （see Refs. [1]—[4]）....Many authors have studied the problem of determining the linear operators, on the n×n matrix algebra M<sub>n</sub>（R） over a commutative R, which preserve idempotent matrices （see Refs. [1]—[4]）. In this note, we make a start on the noncommutative case. If R and R<sub>1</sub> are division rings and their characteristic numbers are not 2 and their centers are the same field F. Let T denote an F-linear operator which maps M<sub>n</sub>（R） into M<sub>n</sub>（R<sub>1</sub>） where M<sub>n</sub>（R）展开更多
文摘Let R be a primitive ring with nonzero socle, M a faithful irreducible right R-module, A the central-izer of M, and L= a direct sum of countably many minimal right ideals L, of R. Then there existsa family of subsetsis infinite) of R such that L=R for any W, whereeach is a set of countably many orthogonal idempotent elements of rank one in R. Furthermore,there exists a primitive ring R and a direct sum L=of countably many minimal right ideals Li ofR, but R has no subset B =of countably many orthogonal idempotent elements of rank one such that and B can be extended to a corresponding basis of some basis of M over A.
文摘The structure theorem of the primitive rings with non-zero socles has been proved in Refs. [1] and (2)In the former, the finite topological method is used; and in the latter the method of right and left double modules. It should be noted that there is a close relation between the structure of the primitive rings of this
文摘It is well known that,for a subring of a full linear ring over a vector spaec,2-foldtransitive implies k-fold transitive for every natual integer k,and a primitive ring withminimal oneside ideal is a two side nonsingular ring and every isomorphism can be inducedby a semi-linear one to one transformation.This paper generalizes these results to weaklyprimitive rings.
文摘In this paper the concept △-context is introduced,as a generalization of MoritaContext,to give a characterization of nonsingularly relative primitive rings.
基金Project supported by the National Natural Science Foundation of China.
文摘Let G be an arbitrary group, finite or infinite, and A be a G-graded ring, i. e. A is an associative ring and A<sub>g</sub> （direct sum of additive subgroups A<sub>g</sub> ）with property: A<sub>g</sub>· A<sub>h</sub> A<sub>gh</sub>, g, h∈G. Let M be a graded A-module, i. e. M is right A-module and M<sub>g</sub>
基金This work is Supported by NSF of Heilongjiang Province
文摘Peal[2] shows that a sufficient and necessary condition on the existence of theMoore-Penrose inverse over any fields.Zhuang [3] generalize the result to any divisionrings.In this section we give another sufficient and necessary condition on the existence ofthe Moore-Penrose inverse over any division rings.Our result can be regarded as an im-provement of Theorem lin[1].As a medium result,we also show a characterization ofthe{1,2}-inverse.
文摘A certain variety of non-switched polynomials provides a uni-figure representation for a wide range of linear functional equations. This is properly adapted for the calculations. We reinterpret from this point of view a number of algorithms.
基金Supported by the National Natural Science Foundation of China(60673081)863 Program(2006AA01Z417)
文摘Let Z/（p^e） be the integer residue ring modulo p^e with p an odd prime and integer e ≥ 3. For a sequence a over Z/（p^e）, there is a unique p-adic decomposition a- = a-0 ＋a-1 .p ＋… ＋ a-e-l .p^e-1 where each a-i can be regarded as a sequence over Z/（p）, 0 ≤ i ≤ e - 1. Let f（x） be a primitive polynomial over Z/（p^e） and G＇（f（x）,p^e） the set of all primitive sequences generated by f（x） over Z/（p^e）. For μ（x） ∈ Z/（p）[x] with deg（μ（x）） ≥ 2 and gad（1 ＋ deg（μ（x））,p- 1） = 1, setφe-1 （x0, x1,… , xe-1） = xe-1. [μ（xe-2） ＋ ηe-3（x0, X1,…, xe-3）] ＋ ηe-2（x0, X1,…, xe-2） which is a function of e variables over Z/（p）. Then the compressing mapφe-1 ： G＇（f（x）,p^e） → （Z/（p））^∞ ,a-→φe-1（a-0,a-1, … ,a-e-1） is injective. That is, for a-,b-∈ G＇（f（x）,p^e）, a- = b- if and only if φe-1 （a-0,a-1, … ,a-e-1） = φe-1（b-0, b-1,… ,b-e-1）. As for the case of e = 2, similar result is also given. Furthermore, if functions φe-1 and ψe-1 over Z/（p） are both of the above form and satisfy φe-1（a-0,a-1,…,a-e-1）=ψe-1（b-0, b-1,… ,b-e-1） for a-,b-∈G＇（f（x）,p^e）, the relations between a- and b-, φe-1 and ψe-1 are discussed
文摘The theory of primitive polynomials over Galois rings is analogue to the same one over finite fields. It also provides useful tools for one to study the maximal period sequences over Galois rings. In the case of F<sub>q</sub>, we have more complete results. In the case of Z<sub>p<sup>n</sup></sub>, n≥2, there are also some results. In particular, according to refs. [3, 4] and using the technique of trace representation of maximal period sequences over F<sub>q</sub>, we have found a discriminant which can judge whether a given polynomial f（x） over Z<sub>p<sup>n</sup></sub> is a primitive polynomial if f（x） mod p is a primitive polynomial over F<sub>p</sub>. Furthermore, it is easy to calculate the discriminant using the coefficients of f（x）.
文摘Many authors have studied the problem of determining the linear operators, on the n×n matrix algebra M<sub>n</sub>（R） over a commutative R, which preserve idempotent matrices （see Refs. [1]—[4]）. In this note, we make a start on the noncommutative case. If R and R<sub>1</sub> are division rings and their characteristic numbers are not 2 and their centers are the same field F. Let T denote an F-linear operator which maps M<sub>n</sub>（R） into M<sub>n</sub>（R<sub>1</sub>） where M<sub>n</sub>（R）
